Based on what I read here:
https://www.matterhornparadise.ch/en/Book/Online-tickets/Matterhorn-Alpine-Crossing
I think you have to purchase the ticket on-site, in person (and not via the website) in order to leverage the Half Fare Card. You purchase it as part of your overall trip from Zermatt. Also, notice that the 50% is only good for getting as far as Testa Gregia, which includes the new Alpine Crossing cable car. I'm pretty sure that If you wish to connect on to the other already existing cable cars to ride into Cervinia, for example, you will need to buy those tickets upon arrival in Teste Grigia.. They are in Italy.
I'm hoping to ride as far as Testa Gregia during my September trip.